Metric geometry
Metric geometry studies metric spaces from a geometrical, rather than analytical, perspective, focusing on intrinsic concepts like geodesic length and curvature.
General
Textbooks
- Burago, Burago, Ivanov, 2001: A course in metric geometry (pdf, errata )
- An accessible introduction and Steele Prize winner
- However, contains many errors, some serious—check the errata!
- Bridson & Haefliger, 1999: Metric spaces of non-positive curvature (doi)
- Part I covers the basic concepts of metric geometry: length spaces, geodesic spaces, the “model spaces” of constant curvature, etc.
- Surprisingly readable and much more carefully written than Burago et al
- On the whole, my preferred introduction to metric geometry
- Papadopoulos, 2014: Metric spaces, convexity and nonpositive curvature,
2nd ed. (doi)
- A careful, systematic book with detailed proofs
- Chapter 3 covers maps between metric spaces, especially nonexpansive maps, drawing on work by Busemann, such as: Busemann, 1964: Length-preserving maps (pdf)
Other books, monographs, and lecture notes
- Gromov, 1991: Metric structures for Riemannian and non-Riemannian spaces
(doi)
- Influential book by Mikhail Gromov, translated from 1981 French original
- Burago et al: “a remarkable book, which gives a vast panorama of ‘geometrical mathematics from a metric viewpoint.’ Unfortunately, Gromov’s book seems hardly accessible to graduate students and non-experts in geometry.”
- Ballmann, 1995: Lectures on spaces of nonpositive curvature (doi, pdf)
- Ballmann, Gromov, Schroeder, 1985: Manifolds of nonpositive curvature (doi)
- Shioya, 2016: Metric measure geometry: Gromov’s theory of convergence and
concentration of metrics and measures (arxiv, doi, pdf)
- Self-described as an expansion on Chapter \(3 \tfrac{1}{2}\) of Gromov’s book
- Alexander, Kapovitch, Petrunin, 2019+, draft: Alexandrov geometry (arxiv)
Special topics
Metric measure spaces
- Villani, 2009: Optimal transport: Old and new, Chapter 27: Convergence of metric-measure spaces (doi)
- Sturm, 2006, in Acta Math.: On the geometry of metric measure spaces, I (doi) and II (doi)
- Sturm, 2012: The space of spaces: Curvature bounds and gradient flows on the
space of metric measure spaces (arxiv)
- Talk: Geometric analysis on the space of metric measure spaces (video)
Metric functionals, aka horofunctions or Busemann functions
- Bridson & Haefliger, 1999, Ch. II.8: The boundary at infinity of a CAT(0) space
- Karlsson, 2019: Elements of a metric spectral theory (arxiv, pdf)
- Proposes metric analogs of linear functionals, eigenvalues, and other elements of linear spectral theory
- Karlsson, 2020: Hahn-Banach for metric functionals and horofunctions (arxiv)
Metric graphs, aka metrized graphs
Metric geometry makes contact with graph theory through metric graphs.
- Burago et al, 2001, Sec. 3.2.2: Metric graphs
- Baker & Faber, 2006: Metrized graphs, Laplacian operators, and electrical
networks (pdf, arxiv)
- Expository paper in a book on quantum graphs (doi)
- Laplacians on metric graphs interpolate between classical Laplacians, which are differential operators, and graph Laplacians
- Baker & Rumely, 2007: Harmonic analysis on metrized graphs (doi, arxiv)
Metric embeddings
Questions about embedding metric spaces into Euclidean spaces and other normed spaces are a staple of discrete and combinatorial geometry, with applications such as kernels in machine learning and the Johnson-Lindenstrauss lemma.
- Deza & Laurent, 1997: Geometry of cuts and metrics (doi, pdf)
- Sec 6.2: “Characterization of \(L_2\)-embeddability” bears on the problem of constructing kernels from metrics, as explained in a blog post by Suresh Venkatasubramanian
- Matoušek, 2002: Lectures on discrete geometry, Chapter 15: Embedding finite metric spaces into normed spaces
- Matoušek, 2013: Lecture notes on metric embeddings (pdf)
- Frequently referenced in Moses Charikar’s 2018 lecture notes for Stanford’s CS 369M: “Metric embeddings and algorithmic applications”
See also Harald Räcke’s 2006 lecture notes on metric embeddings.
Miscellany
- Tuzhilin, 2016: Who invented the Gromov-Hausdorff distance? (arxiv)
- Argues that mathematician David Edwards deserves some of the credit for the Gromov-Hausdorff distance
- Edwards, 1975: The Structure of Superspace (doi, pdf)
- Existence of measures invariant under isometries (Math.SE )